Here is the definition of principal logarithm. Let $Y: F_1\to F_2$ be a bounded operator on a complex Banach space $F_1$, and denote $\mathscr{S}(Y)=\{B|\exp ~B = Y\}$. There exists a unique member $A$ of $\mathscr{S}(Y)$ satisfying $A\in D_2 = \{B:F_1\to F_2|~\sigma(B)\subset\{\lambda:|\text{Im}(\lambda)|<\pi\}\}$, corresponding to the principal logarithm of $Y$, where $\sigma(A)$ denote the spectrum of $A$. Then we define the principal logarithm $\text{Log}(Y)=A$.
Here is the problem. Consider $Y\in D_1=\{\exp(B):\sigma(B)\subset\{\lambda:|\text{Im}(\lambda)|<\pi\}\}$ and the principal logarithm $\text{Log}:D_1\to D_2$. How to prove the continuity of the principal logarithm function $\text{Log}(Y)$? That is to prove, $\forall \epsilon>0, \exists \delta>0, s.t. \|(\text{Log}~Y_1 - \text{Log}~Y_2)x\|<\epsilon, \forall x\in X,$ if $\|(Y_1-Y_2)x\|<\delta$. Or, is there the continuity proof of other logarithm definition of linear operators?
ps: I think it is intuitive because principal logarithm Log is kind of inverse function of continuous function $\exp$ for $A\in D_2$, i.e., $\text{Log}(\exp(A)) = A$. But I do not know how to prove it. I would be really thankful if you help me a little.